Factoring

MathStudent1999

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Mar 18, 2012
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How do I find the greatest prime factor of a number like 2^26 + 1? For -1 I can use the difference of squares but what can I do for plus one?
 
How do I find the greatest prime factor of a number like 2^26 + 1?
For -1 I know you are referring to \(\displaystyle 2^{26} - 1\) with that.
I can use the difference of squares but what can I do for plus one?

MathStudent1999,consider that it can be rewritten as:

\(\displaystyle (2^2)^{13} + 1 \ = \)

\(\displaystyle 4^{13} + 1\)

At the moment, I don't know about the greatest prime factor, but because it is the sum of two odd powers
(the degree is > = 3), it can be factored as:


\(\displaystyle (4 + 1)(4^{12} - 4^{11} + 4^{10} - 4^9 + \ . . . \ + 4^2 - 4^1 + 1)\)


At least there is a prime factor of 5.
 
Another way to get 5 as a factor would be as follows:

The unit digit of 2n - as n goes from 1 → n would be 2,4,8,6,2,4,8....

26 = 6*4+2

Hence the unit digit of 226 is 4

When we add 1 to 226, the unit digit becomes 5. Hence 226+1 is divisible by 5.
 
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